# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Ascent spectrum and essential ascent spectrum

### Tom 187 / 2008

Studia Mathematica 187 (2008), 59-73 MSC: 47A53, 47A55. DOI: 10.4064/sm187-1-3

#### Streszczenie

We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space $X$ has finite dimension if and only if the essential ascent of every operator on $X$ is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator $F$ on $X$ has some finite rank power if and only if $\sigma_{{\rm asc}}^{{\rm e}} (T+F)=\sigma_{{\rm asc}}^{{\rm e}} (T)$ for every operator $T$ commuting with $F$. The quasi-nilpotent part, the analytic core and the single-valued extension property are also analyzed for operators with finite essential ascent.

#### Autorzy

• O. Bel Hadj FredjUFR de Mathématiques, UMR-CNRS 8524
Université de Lille 1
59655 Villeneuve d'Ascq, France
e-mail
• M. BurgosDepartamento de Análisis Matemático